Properties

Label 5850.2.e.e.5149.1
Level $5850$
Weight $2$
Character 5850.5149
Analytic conductor $46.712$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5850.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(46.7124851824\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 5149.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 5850.5149
Dual form 5850.2.e.e.5149.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{8} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{8} -4.00000 q^{11} -1.00000i q^{13} +1.00000 q^{16} +6.00000i q^{17} -4.00000 q^{19} +4.00000i q^{22} +8.00000i q^{23} -1.00000 q^{26} +6.00000 q^{29} -8.00000 q^{31} -1.00000i q^{32} +6.00000 q^{34} -10.0000i q^{37} +4.00000i q^{38} +6.00000 q^{41} -4.00000i q^{43} +4.00000 q^{44} +8.00000 q^{46} +7.00000 q^{49} +1.00000i q^{52} -10.0000i q^{53} -6.00000i q^{58} +4.00000 q^{59} -2.00000 q^{61} +8.00000i q^{62} -1.00000 q^{64} -12.0000i q^{67} -6.00000i q^{68} -16.0000 q^{71} -2.00000i q^{73} -10.0000 q^{74} +4.00000 q^{76} +16.0000 q^{79} -6.00000i q^{82} -12.0000i q^{83} -4.00000 q^{86} -4.00000i q^{88} +10.0000 q^{89} -8.00000i q^{92} -6.00000i q^{97} -7.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + O(q^{10}) \) \( 2q - 2q^{4} - 8q^{11} + 2q^{16} - 8q^{19} - 2q^{26} + 12q^{29} - 16q^{31} + 12q^{34} + 12q^{41} + 8q^{44} + 16q^{46} + 14q^{49} + 8q^{59} - 4q^{61} - 2q^{64} - 32q^{71} - 20q^{74} + 8q^{76} + 32q^{79} - 8q^{86} + 20q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5850\mathbb{Z}\right)^\times\).

\(n\) \(2251\) \(3251\) \(3277\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) − 1.00000i − 0.277350i
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000i 1.45521i 0.685994 + 0.727607i \(0.259367\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 4.00000i 0.852803i
\(23\) 8.00000i 1.66812i 0.551677 + 0.834058i \(0.313988\pi\)
−0.551677 + 0.834058i \(0.686012\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) 0 0
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 0 0
\(37\) − 10.0000i − 1.64399i −0.569495 0.821995i \(-0.692861\pi\)
0.569495 0.821995i \(-0.307139\pi\)
\(38\) 4.00000i 0.648886i
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 1.00000i 0.138675i
\(53\) − 10.0000i − 1.37361i −0.726844 0.686803i \(-0.759014\pi\)
0.726844 0.686803i \(-0.240986\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) − 6.00000i − 0.787839i
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 8.00000i 1.01600i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 12.0000i − 1.46603i −0.680211 0.733017i \(-0.738112\pi\)
0.680211 0.733017i \(-0.261888\pi\)
\(68\) − 6.00000i − 0.727607i
\(69\) 0 0
\(70\) 0 0
\(71\) −16.0000 −1.89885 −0.949425 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) 0 0
\(73\) − 2.00000i − 0.234082i −0.993127 0.117041i \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) −10.0000 −1.16248
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) 16.0000 1.80014 0.900070 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 6.00000i − 0.662589i
\(83\) − 12.0000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) − 4.00000i − 0.426401i
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 8.00000i − 0.834058i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 6.00000i − 0.609208i −0.952479 0.304604i \(-0.901476\pi\)
0.952479 0.304604i \(-0.0985241\pi\)
\(98\) − 7.00000i − 0.707107i
\(99\) 0 0
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 10.0000i 0.940721i 0.882474 + 0.470360i \(0.155876\pi\)
−0.882474 + 0.470360i \(0.844124\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) − 4.00000i − 0.368230i
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 2.00000i 0.181071i
\(123\) 0 0
\(124\) 8.00000 0.718421
\(125\) 0 0
\(126\) 0 0
\(127\) 8.00000i 0.709885i 0.934888 + 0.354943i \(0.115500\pi\)
−0.934888 + 0.354943i \(0.884500\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) − 10.0000i − 0.854358i −0.904167 0.427179i \(-0.859507\pi\)
0.904167 0.427179i \(-0.140493\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 16.0000i 1.34269i
\(143\) 4.00000i 0.334497i
\(144\) 0 0
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) 0 0
\(148\) 10.0000i 0.821995i
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) − 4.00000i − 0.324443i
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 2.00000i − 0.159617i −0.996810 0.0798087i \(-0.974569\pi\)
0.996810 0.0798087i \(-0.0254309\pi\)
\(158\) − 16.0000i − 1.27289i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 4.00000i − 0.313304i −0.987654 0.156652i \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) − 8.00000i − 0.619059i −0.950890 0.309529i \(-0.899829\pi\)
0.950890 0.309529i \(-0.100171\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 4.00000i 0.304997i
\(173\) − 18.0000i − 1.36851i −0.729241 0.684257i \(-0.760127\pi\)
0.729241 0.684257i \(-0.239873\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) − 10.0000i − 0.749532i
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −8.00000 −0.589768
\(185\) 0 0
\(186\) 0 0
\(187\) − 24.0000i − 1.75505i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) − 10.0000i − 0.719816i −0.932988 0.359908i \(-0.882808\pi\)
0.932988 0.359908i \(-0.117192\pi\)
\(194\) −6.00000 −0.430775
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) − 6.00000i − 0.427482i −0.976890 0.213741i \(-0.931435\pi\)
0.976890 0.213741i \(-0.0685649\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 2.00000i − 0.140720i
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) − 1.00000i − 0.0693375i
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 10.0000i 0.686803i
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) − 10.0000i − 0.677285i
\(219\) 0 0
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 0 0
\(223\) − 8.00000i − 0.535720i −0.963458 0.267860i \(-0.913684\pi\)
0.963458 0.267860i \(-0.0863164\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 10.0000 0.665190
\(227\) − 4.00000i − 0.265489i −0.991150 0.132745i \(-0.957621\pi\)
0.991150 0.132745i \(-0.0423790\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.00000i 0.393919i
\(233\) − 14.0000i − 0.917170i −0.888650 0.458585i \(-0.848356\pi\)
0.888650 0.458585i \(-0.151644\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −4.00000 −0.260378
\(237\) 0 0
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) − 5.00000i − 0.321412i
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) 4.00000i 0.254514i
\(248\) − 8.00000i − 0.508001i
\(249\) 0 0
\(250\) 0 0
\(251\) 28.0000 1.76734 0.883672 0.468106i \(-0.155064\pi\)
0.883672 + 0.468106i \(0.155064\pi\)
\(252\) 0 0
\(253\) − 32.0000i − 2.01182i
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.00000i 0.374270i 0.982334 + 0.187135i \(0.0599201\pi\)
−0.982334 + 0.187135i \(0.940080\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 12.0000i 0.741362i
\(263\) 24.0000i 1.47990i 0.672660 + 0.739952i \(0.265152\pi\)
−0.672660 + 0.739952i \(0.734848\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 12.0000i 0.733017i
\(269\) −26.0000 −1.58525 −0.792624 0.609711i \(-0.791286\pi\)
−0.792624 + 0.609711i \(0.791286\pi\)
\(270\) 0 0
\(271\) 24.0000 1.45790 0.728948 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) 6.00000i 0.363803i
\(273\) 0 0
\(274\) −10.0000 −0.604122
\(275\) 0 0
\(276\) 0 0
\(277\) − 10.0000i − 0.600842i −0.953807 0.300421i \(-0.902873\pi\)
0.953807 0.300421i \(-0.0971271\pi\)
\(278\) 12.0000i 0.719712i
\(279\) 0 0
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) − 20.0000i − 1.18888i −0.804141 0.594438i \(-0.797374\pi\)
0.804141 0.594438i \(-0.202626\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) 0 0
\(288\) 0 0
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 2.00000i 0.117041i
\(293\) − 26.0000i − 1.51894i −0.650545 0.759468i \(-0.725459\pi\)
0.650545 0.759468i \(-0.274541\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 10.0000 0.581238
\(297\) 0 0
\(298\) − 6.00000i − 0.347571i
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) 0 0
\(307\) − 12.0000i − 0.684876i −0.939540 0.342438i \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) 22.0000i 1.24351i 0.783210 + 0.621757i \(0.213581\pi\)
−0.783210 + 0.621757i \(0.786419\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) 2.00000i 0.112331i 0.998421 + 0.0561656i \(0.0178875\pi\)
−0.998421 + 0.0561656i \(0.982113\pi\)
\(318\) 0 0
\(319\) −24.0000 −1.34374
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 24.0000i − 1.33540i
\(324\) 0 0
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) 0 0
\(328\) 6.00000i 0.331295i
\(329\) 0 0
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 12.0000i 0.658586i
\(333\) 0 0
\(334\) −8.00000 −0.437741
\(335\) 0 0
\(336\) 0 0
\(337\) 2.00000i 0.108947i 0.998515 + 0.0544735i \(0.0173480\pi\)
−0.998515 + 0.0544735i \(0.982652\pi\)
\(338\) 1.00000i 0.0543928i
\(339\) 0 0
\(340\) 0 0
\(341\) 32.0000 1.73290
\(342\) 0 0
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) − 20.0000i − 1.07366i −0.843692 0.536828i \(-0.819622\pi\)
0.843692 0.536828i \(-0.180378\pi\)
\(348\) 0 0
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 4.00000i 0.213201i
\(353\) − 30.0000i − 1.59674i −0.602168 0.798369i \(-0.705696\pi\)
0.602168 0.798369i \(-0.294304\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) − 12.0000i − 0.634220i
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) − 6.00000i − 0.315353i
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 24.0000i − 1.25279i −0.779506 0.626395i \(-0.784530\pi\)
0.779506 0.626395i \(-0.215470\pi\)
\(368\) 8.00000i 0.417029i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) − 22.0000i − 1.13912i −0.821951 0.569558i \(-0.807114\pi\)
0.821951 0.569558i \(-0.192886\pi\)
\(374\) −24.0000 −1.24101
\(375\) 0 0
\(376\) 0 0
\(377\) − 6.00000i − 0.309016i
\(378\) 0 0
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 32.0000i 1.63512i 0.575841 + 0.817562i \(0.304675\pi\)
−0.575841 + 0.817562i \(0.695325\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) 0 0
\(388\) 6.00000i 0.304604i
\(389\) −34.0000 −1.72387 −0.861934 0.507020i \(-0.830747\pi\)
−0.861934 + 0.507020i \(0.830747\pi\)
\(390\) 0 0
\(391\) −48.0000 −2.42746
\(392\) 7.00000i 0.353553i
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 0 0
\(397\) 30.0000i 1.50566i 0.658217 + 0.752828i \(0.271311\pi\)
−0.658217 + 0.752828i \(0.728689\pi\)
\(398\) − 8.00000i − 0.401004i
\(399\) 0 0
\(400\) 0 0
\(401\) −34.0000 −1.69788 −0.848939 0.528490i \(-0.822758\pi\)
−0.848939 + 0.528490i \(0.822758\pi\)
\(402\) 0 0
\(403\) 8.00000i 0.398508i
\(404\) −2.00000 −0.0995037
\(405\) 0 0
\(406\) 0 0
\(407\) 40.0000i 1.98273i
\(408\) 0 0
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −1.00000 −0.0490290
\(417\) 0 0
\(418\) − 16.0000i − 0.782586i
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) 0 0
\(421\) −34.0000 −1.65706 −0.828529 0.559946i \(-0.810822\pi\)
−0.828529 + 0.559946i \(0.810822\pi\)
\(422\) 12.0000i 0.584151i
\(423\) 0 0
\(424\) 10.0000 0.485643
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) − 12.0000i − 0.580042i
\(429\) 0 0
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 0 0
\(433\) − 34.0000i − 1.63394i −0.576683 0.816968i \(-0.695653\pi\)
0.576683 0.816968i \(-0.304347\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) − 32.0000i − 1.53077i
\(438\) 0 0
\(439\) 40.0000 1.90910 0.954548 0.298057i \(-0.0963387\pi\)
0.954548 + 0.298057i \(0.0963387\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 6.00000i − 0.285391i
\(443\) 36.0000i 1.71041i 0.518289 + 0.855206i \(0.326569\pi\)
−0.518289 + 0.855206i \(0.673431\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) 0 0
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) −24.0000 −1.13012
\(452\) − 10.0000i − 0.470360i
\(453\) 0 0
\(454\) −4.00000 −0.187729
\(455\) 0 0
\(456\) 0 0
\(457\) 34.0000i 1.59045i 0.606313 + 0.795226i \(0.292648\pi\)
−0.606313 + 0.795226i \(0.707352\pi\)
\(458\) 14.0000i 0.654177i
\(459\) 0 0
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) − 8.00000i − 0.371792i −0.982569 0.185896i \(-0.940481\pi\)
0.982569 0.185896i \(-0.0595187\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) −14.0000 −0.648537
\(467\) − 12.0000i − 0.555294i −0.960683 0.277647i \(-0.910445\pi\)
0.960683 0.277647i \(-0.0895545\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 4.00000i 0.184115i
\(473\) 16.0000i 0.735681i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 8.00000i 0.365911i
\(479\) −8.00000 −0.365529 −0.182765 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) 0 0
\(481\) −10.0000 −0.455961
\(482\) − 18.0000i − 0.819878i
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) 0 0
\(487\) − 16.0000i − 0.725029i −0.931978 0.362515i \(-0.881918\pi\)
0.931978 0.362515i \(-0.118082\pi\)
\(488\) − 2.00000i − 0.0905357i
\(489\) 0 0
\(490\) 0 0
\(491\) −4.00000 −0.180517 −0.0902587 0.995918i \(-0.528769\pi\)
−0.0902587 + 0.995918i \(0.528769\pi\)
\(492\) 0 0
\(493\) 36.0000i 1.62136i
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 28.0000i − 1.24970i
\(503\) 8.00000i 0.356702i 0.983967 + 0.178351i \(0.0570763\pi\)
−0.983967 + 0.178351i \(0.942924\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −32.0000 −1.42257
\(507\) 0 0
\(508\) − 8.00000i − 0.354943i
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 1.00000i − 0.0441942i
\(513\) 0 0
\(514\) 6.00000 0.264649
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) − 36.0000i − 1.57417i −0.616844 0.787085i \(-0.711589\pi\)
0.616844 0.787085i \(-0.288411\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) − 48.0000i − 2.09091i
\(528\) 0 0
\(529\) −41.0000 −1.78261
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 6.00000i − 0.259889i
\(534\) 0 0
\(535\) 0 0
\(536\) 12.0000 0.518321
\(537\) 0 0
\(538\) 26.0000i 1.12094i
\(539\) −28.0000 −1.20605
\(540\) 0 0
\(541\) 22.0000 0.945854 0.472927 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) − 24.0000i − 1.03089i
\(543\) 0 0
\(544\) 6.00000 0.257248
\(545\) 0 0
\(546\) 0 0
\(547\) − 20.0000i − 0.855138i −0.903983 0.427569i \(-0.859370\pi\)
0.903983 0.427569i \(-0.140630\pi\)
\(548\) 10.0000i 0.427179i
\(549\) 0 0
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) 0 0
\(554\) −10.0000 −0.424859
\(555\) 0 0
\(556\) 12.0000 0.508913
\(557\) 18.0000i 0.762684i 0.924434 + 0.381342i \(0.124538\pi\)
−0.924434 + 0.381342i \(0.875462\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) 10.0000i 0.421825i
\(563\) 12.0000i 0.505740i 0.967500 + 0.252870i \(0.0813744\pi\)
−0.967500 + 0.252870i \(0.918626\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −20.0000 −0.840663
\(567\) 0 0
\(568\) − 16.0000i − 0.671345i
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) − 4.00000i − 0.167248i
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 42.0000i 1.74848i 0.485491 + 0.874241i \(0.338641\pi\)
−0.485491 + 0.874241i \(0.661359\pi\)
\(578\) 19.0000i 0.790296i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 40.0000i 1.65663i
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) −26.0000 −1.07405
\(587\) − 12.0000i − 0.495293i −0.968850 0.247647i \(-0.920343\pi\)
0.968850 0.247647i \(-0.0796572\pi\)
\(588\) 0 0
\(589\) 32.0000 1.31854
\(590\) 0 0
\(591\) 0 0
\(592\) − 10.0000i − 0.410997i
\(593\) 34.0000i 1.39621i 0.715994 + 0.698106i \(0.245974\pi\)
−0.715994 + 0.698106i \(0.754026\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 0 0
\(598\) − 8.00000i − 0.327144i
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −6.00000 −0.244745 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 8.00000i − 0.324710i −0.986732 0.162355i \(-0.948091\pi\)
0.986732 0.162355i \(-0.0519090\pi\)
\(608\) 4.00000i 0.162221i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 10.0000i 0.403896i 0.979396 + 0.201948i \(0.0647272\pi\)
−0.979396 + 0.201948i \(0.935273\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) 0 0
\(617\) − 42.0000i − 1.69086i −0.534089 0.845428i \(-0.679345\pi\)
0.534089 0.845428i \(-0.320655\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 24.0000i − 0.962312i
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 22.0000 0.879297
\(627\) 0 0
\(628\) 2.00000i 0.0798087i
\(629\) 60.0000 2.39236
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 16.0000i 0.636446i
\(633\) 0 0
\(634\) 2.00000 0.0794301
\(635\) 0 0
\(636\) 0 0
\(637\) − 7.00000i − 0.277350i
\(638\) 24.0000i 0.950169i
\(639\) 0 0
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 0 0
\(643\) − 4.00000i − 0.157745i −0.996885 0.0788723i \(-0.974868\pi\)
0.996885 0.0788723i \(-0.0251319\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) − 24.0000i − 0.943537i −0.881722 0.471769i \(-0.843616\pi\)
0.881722 0.471769i \(-0.156384\pi\)
\(648\) 0 0
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) 4.00000i 0.156652i
\(653\) − 34.0000i − 1.33052i −0.746611 0.665261i \(-0.768320\pi\)
0.746611 0.665261i \(-0.231680\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) 0 0
\(658\) 0 0
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 20.0000i 0.777322i
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) 0 0
\(667\) 48.0000i 1.85857i
\(668\) 8.00000i 0.309529i
\(669\) 0 0
\(670\) 0 0
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) − 34.0000i − 1.31060i −0.755367 0.655302i \(-0.772541\pi\)
0.755367 0.655302i \(-0.227459\pi\)
\(674\) 2.00000 0.0770371
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) − 22.0000i − 0.845529i −0.906240 0.422764i \(-0.861060\pi\)
0.906240 0.422764i \(-0.138940\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) − 32.0000i − 1.22534i
\(683\) 28.0000i 1.07139i 0.844411 + 0.535695i \(0.179950\pi\)
−0.844411 + 0.535695i \(0.820050\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) − 4.00000i − 0.152499i
\(689\) −10.0000 −0.380970
\(690\) 0 0
\(691\) −12.0000 −0.456502 −0.228251 0.973602i \(-0.573301\pi\)
−0.228251 + 0.973602i \(0.573301\pi\)
\(692\) 18.0000i 0.684257i
\(693\) 0 0
\(694\) −20.0000 −0.759190
\(695\) 0 0
\(696\) 0 0
\(697\) 36.0000i 1.36360i
\(698\) 6.00000i 0.227103i
\(699\) 0 0
\(700\) 0 0
\(701\) −22.0000 −0.830929 −0.415464 0.909610i \(-0.636381\pi\)
−0.415464 + 0.909610i \(0.636381\pi\)
\(702\) 0 0
\(703\) 40.0000i 1.50863i
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) −30.0000 −1.12906
\(707\) 0 0
\(708\) 0 0
\(709\) 50.0000 1.87779 0.938895 0.344204i \(-0.111851\pi\)
0.938895 + 0.344204i \(0.111851\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 10.0000i 0.374766i
\(713\) − 64.0000i − 2.39682i
\(714\) 0 0
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) − 16.0000i − 0.597115i
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 3.00000i 0.111648i
\(723\) 0 0
\(724\) −6.00000 −0.222988
\(725\) 0 0
\(726\) 0 0
\(727\) − 16.0000i − 0.593407i −0.954970 0.296704i \(-0.904113\pi\)
0.954970 0.296704i \(-0.0958873\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) 0 0
\(733\) 50.0000i 1.84679i 0.383849 + 0.923396i \(0.374598\pi\)
−0.383849 + 0.923396i \(0.625402\pi\)
\(734\) −24.0000 −0.885856
\(735\) 0 0
\(736\) 8.00000 0.294884
\(737\) 48.0000i 1.76810i
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 24.0000i − 0.880475i −0.897881 0.440237i \(-0.854894\pi\)
0.897881 0.440237i \(-0.145106\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −22.0000 −0.805477
\(747\) 0 0
\(748\) 24.0000i 0.877527i
\(749\) 0 0
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −6.00000 −0.218507
\(755\) 0 0
\(756\) 0 0
\(757\) − 42.0000i − 1.52652i −0.646094 0.763258i \(-0.723599\pi\)
0.646094 0.763258i \(-0.276401\pi\)
\(758\) − 4.00000i − 0.145287i
\(759\) 0 0
\(760\) 0 0
\(761\) −42.0000 −1.52250 −0.761249 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 32.0000 1.15621
\(767\) − 4.00000i − 0.144432i
\(768\) 0 0
\(769\) −34.0000 −1.22607 −0.613036 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 10.0000i 0.359908i
\(773\) 38.0000i 1.36677i 0.730061 + 0.683383i \(0.239492\pi\)
−0.730061 + 0.683383i \(0.760508\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 6.00000 0.215387
\(777\) 0 0
\(778\) 34.0000i 1.21896i
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 64.0000 2.29010
\(782\) 48.0000i 1.71648i
\(783\) 0 0
\(784\) 7.00000 0.250000
\(785\) 0 0
\(786\) 0 0
\(787\) 4.00000i 0.142585i 0.997455 + 0.0712923i \(0.0227123\pi\)
−0.997455 + 0.0712923i \(0.977288\pi\)
\(788\) 6.00000i 0.213741i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2.00000i 0.0710221i
\(794\) 30.0000 1.06466
\(795\) 0 0
\(796\) −8.00000 −0.283552
\(797\) 2.00000i 0.0708436i 0.999372 + 0.0354218i \(0.0112775\pi\)
−0.999372 + 0.0354218i \(0.988723\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 34.0000i 1.20058i
\(803\) 8.00000i 0.282314i
\(804\) 0 0
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) 0 0
\(808\) 2.00000i 0.0703598i
\(809\) 42.0000 1.47664 0.738321 0.674450i \(-0.235619\pi\)
0.738321 + 0.674450i \(0.235619\pi\)
\(810\) 0 0
\(811\) 12.0000 0.421377 0.210688 0.977553i \(-0.432429\pi\)
0.210688 + 0.977553i \(0.432429\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 40.0000 1.40200
\(815\) 0 0
\(816\) 0 0
\(817\) 16.0000i 0.559769i
\(818\) − 22.0000i − 0.769212i
\(819\) 0 0
\(820\) 0 0
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) 0 0
\(823\) − 16.0000i − 0.557725i −0.960331 0.278862i \(-0.910043\pi\)
0.960331 0.278862i \(-0.0899574\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 36.0000i 1.25184i 0.779886 + 0.625921i \(0.215277\pi\)
−0.779886 + 0.625921i \(0.784723\pi\)
\(828\) 0 0
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.00000i 0.0346688i
\(833\) 42.0000i 1.45521i
\(834\) 0 0
\(835\) 0 0
\(836\) −16.0000 −0.553372
\(837\) 0 0
\(838\) 4.00000i 0.138178i
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 34.0000i 1.17172i
\(843\) 0 0
\(844\) 12.0000 0.413057
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) − 10.0000i − 0.343401i
\(849\) 0 0
\(850\) 0 0
\(851\) 80.0000 2.74236
\(852\) 0 0
\(853\) 26.0000i 0.890223i 0.895475 + 0.445112i \(0.146836\pi\)
−0.895475 + 0.445112i \(0.853164\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) − 18.0000i − 0.614868i −0.951569 0.307434i \(-0.900530\pi\)
0.951569 0.307434i \(-0.0994704\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 24.0000i − 0.817443i
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −34.0000 −1.15537
\(867\) 0 0
\(868\) 0 0
\(869\) −64.0000 −2.17105
\(870\) 0 0
\(871\) −12.0000 −0.406604
\(872\) 10.0000i 0.338643i
\(873\) 0 0
\(874\) −32.0000 −1.08242
\(875\) 0 0
\(876\) 0 0
\(877\) 14.0000i 0.472746i 0.971662 + 0.236373i \(0.0759588\pi\)
−0.971662 + 0.236373i \(0.924041\pi\)
\(878\) − 40.0000i − 1.34993i
\(879\) 0 0
\(880\) 0 0
\(881\) −2.00000 −0.0673817 −0.0336909 0.999432i \(-0.510726\pi\)
−0.0336909 + 0.999432i \(0.510726\pi\)
\(882\) 0 0
\(883\) 36.0000i 1.21150i 0.795656 + 0.605748i \(0.207126\pi\)
−0.795656 + 0.605748i \(0.792874\pi\)
\(884\) −6.00000 −0.201802
\(885\) 0 0
\(886\) 36.0000 1.20944
\(887\) − 8.00000i − 0.268614i −0.990940 0.134307i \(-0.957119\pi\)
0.990940 0.134307i \(-0.0428808\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 8.00000i 0.267860i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) − 18.0000i − 0.600668i
\(899\) −48.0000 −1.60089
\(900\) 0 0
\(901\) 60.0000 1.99889
\(902\) 24.0000i 0.799113i
\(903\) 0 0
\(904\) −10.0000 −0.332595
\(905\) 0 0
\(906\) 0 0
\(907\) − 12.0000i − 0.398453i −0.979953 0.199227i \(-0.936157\pi\)
0.979953 0.199227i \(-0.0638430\pi\)
\(908\) 4.00000i 0.132745i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 48.0000i 1.58857i
\(914\) 34.0000 1.12462
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) 0 0
\(918\) 0 0
\(919\) 56.0000 1.84727 0.923635 0.383274i \(-0.125203\pi\)
0.923635 + 0.383274i \(0.125203\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 18.0000i − 0.592798i
\(923\) 16.0000i 0.526646i
\(924\) 0 0
\(925\) 0 0
\(926\) −8.00000 −0.262896
\(927\) 0 0
\(928\) − 6.00000i − 0.196960i
\(929\) −14.0000 −0.459325 −0.229663 0.973270i \(-0.573762\pi\)
−0.229663 + 0.973270i \(0.573762\pi\)
\(930\) 0 0
\(931\) −28.0000 −0.917663
\(932\) 14.0000i 0.458585i
\(933\) 0 0
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) 0 0
\(937\) − 22.0000i − 0.718709i −0.933201 0.359354i \(-0.882997\pi\)
0.933201 0.359354i \(-0.117003\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −30.0000 −0.977972 −0.488986 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) 0 0
\(943\) 48.0000i 1.56310i
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) 16.0000 0.520205
\(947\) − 52.0000i − 1.68977i −0.534946 0.844886i \(-0.679668\pi\)
0.534946 0.844886i \(-0.320332\pi\)
\(948\) 0 0
\(949\) −2.00000 −0.0649227
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 34.0000i 1.10137i 0.834714 + 0.550684i \(0.185633\pi\)
−0.834714 + 0.550684i \(0.814367\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 8.00000 0.258738
\(957\) 0 0
\(958\) 8.00000i 0.258468i
\(959\) 0 0
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 10.0000i 0.322413i
\(963\) 0 0
\(964\) −18.0000 −0.579741
\(965\) 0 0
\(966\) 0 0
\(967\) 16.0000i 0.514525i 0.966342 + 0.257263i \(0.0828206\pi\)
−0.966342 + 0.257263i \(0.917179\pi\)
\(968\) 5.00000i 0.160706i
\(969\) 0 0
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) 30.0000i 0.959785i 0.877327 + 0.479893i \(0.159324\pi\)
−0.877327 + 0.479893i \(0.840676\pi\)
\(978\) 0 0
\(979\) −40.0000 −1.27841
\(980\) 0 0
\(981\) 0 0
\(982\) 4.00000i 0.127645i
\(983\) 8.00000i 0.255160i 0.991828 + 0.127580i \(0.0407210\pi\)
−0.991828 + 0.127580i \(0.959279\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 36.0000 1.14647
\(987\) 0 0
\(988\) − 4.00000i − 0.127257i
\(989\) 32.0000 1.01754
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 8.00000i 0.254000i
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 58.0000i − 1.83688i −0.395562 0.918439i \(-0.629450\pi\)
0.395562 0.918439i \(-0.370550\pi\)
\(998\) 20.0000i 0.633089i
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 5850.2.e.e.5149.1 2
3.2 odd 2 1950.2.e.g.1249.2 2
5.2 odd 4 5850.2.a.bo.1.1 1
5.3 odd 4 1170.2.a.a.1.1 1
5.4 even 2 inner 5850.2.e.e.5149.2 2
15.2 even 4 1950.2.a.k.1.1 1
15.8 even 4 390.2.a.f.1.1 1
15.14 odd 2 1950.2.e.g.1249.1 2
20.3 even 4 9360.2.a.p.1.1 1
60.23 odd 4 3120.2.a.w.1.1 1
195.8 odd 4 5070.2.b.d.1351.2 2
195.38 even 4 5070.2.a.a.1.1 1
195.83 odd 4 5070.2.b.d.1351.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.f.1.1 1 15.8 even 4
1170.2.a.a.1.1 1 5.3 odd 4
1950.2.a.k.1.1 1 15.2 even 4
1950.2.e.g.1249.1 2 15.14 odd 2
1950.2.e.g.1249.2 2 3.2 odd 2
3120.2.a.w.1.1 1 60.23 odd 4
5070.2.a.a.1.1 1 195.38 even 4
5070.2.b.d.1351.1 2 195.83 odd 4
5070.2.b.d.1351.2 2 195.8 odd 4
5850.2.a.bo.1.1 1 5.2 odd 4
5850.2.e.e.5149.1 2 1.1 even 1 trivial
5850.2.e.e.5149.2 2 5.4 even 2 inner
9360.2.a.p.1.1 1 20.3 even 4